In Table REF , we give the integrated rates of the asymmetries shown in Fig. REF , i.e., the values of \(A_\xi \) . Comparing with the values of \(A_\xi \sim 10^{-4}\)  for the \(K_{l3\gamma }\)  decays [1]}, [2]}, [3]}), the corresponding results of the T-odd asymmetries in the \(\tau \rightarrow \pi \pi \gamma \nu _\tau \)  are around two orders of magnitudes larger, and hence are very encouraging for the future experimental measurements conducted in Belle-II [4]} and the prospective super tau-charm facility [5]}. Moreover, different amplitudes with different resonance couplings lead to obviously different shapes of the T-odd asymmetry distributions with respect to the \(\xi \)  variable. Therefore the precise measurements of the T-odd asymmetry distributions as functions of \(\xi \)  also provide another quantity to discern different resonance dynamics in the \(\tau \rightarrow \pi \pi \gamma \nu _\tau \)  process.
